What is the derivative of xe^{(x^2/2}) ?

The derivative of xe^{(x^2/2}) is e^{(x^2)/2}+e^{(x^2)/2}x^2

What is the first derivative of xe^{(x^2/2}) ?

The first derivative of xe^{(x^2/2}) is e^{(x^2)/2}+e^{(x^2)/2}x^2

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\frac{d}{d x} (x e^{\frac{x ^{2}}{2}})

e^{\frac{x ^{2}}{2}} + e^{\frac{x ^{2}}{2}} x^{2}

Solution steps

\frac{d}{d x} (x e^{\frac{x ^{2}}{2}})

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x, g = e^{\frac{x ^{2}}{2}}

= \frac{d x}{d x} e^{\frac{x ^{2}}{2}} + \frac{d}{d x} (e^{\frac{x ^{2}}{2}}) x

\frac{d x}{d x} = 1

\frac{d}{d x} (e^{\frac{x ^{2}}{2}}) = e^{\frac{x ^{2}}{2}} x

= 1 \cdot e^{\frac{x ^{2}}{2}} + e^{\frac{x ^{2}}{2}} xx

Simplify

1 \cdot e^{\frac{x ^{2}}{2}} + e^{\frac{x ^{2}}{2}} xx : e^{\frac{x ^{2}}{2}} + e^{\frac{x ^{2}}{2}} x^{2}

= e^{\frac{x ^{2}}{2}} + e^{\frac{x ^{2}}{2}} x^{2}

\int \frac{3 + 4 x}{1 + x ^{2}} d x

\int_{0}^{\frac{π}{4}} x^{2} cos (2 x) d x

f (x) = cos (8 x)

y^{^{''}} + 5 y = 0, y (0) = 1, y^{^{'}} (0) = 1

\frac{\partial}{\partial x} (cos (2 y - x))

What is the derivative of xe^{(x^2/2}) ?
The derivative of xe^{(x^2/2}) is e^{(x^2)/2}+e^{(x^2)/2}x^2
What is the first derivative of xe^{(x^2/2}) ?
The first derivative of xe^{(x^2/2}) is e^{(x^2)/2}+e^{(x^2)/2}x^2